Bill: The existence of Hamel bases in (all) vector spaces is equivalent to the axiom of choice (see Blass, Andreas "Existence of bases implies the axiom of choice". Contemporary mathematics 31, 1984). Also, it is consistent with ZF that all linear functionals on $l_2$ are continuous (for instance, a model in which every set of reals has the property of Baire).

Bill: The existence of Hamel basis in (all) vector spaces is equivalent to the axiom of choice (see Blass, Andreas "Existence of bases implies the axiom of choice". Contemporary mathematics 31, 1984). For the existence of a Hamel basis in $l_2$ it is enough to have a well-ordering of the reals. Also, it is consistent with ZF that all linear functionals on $l_2$ are continuous (for instance, a model in which every set of reals has the property of Baire).

Bill: The existence of Hamel bases in (all) vector spaces is equivalent to the axiom of choice. Also, it is consistent with ZF that all linear functionals on $l_2$ are continuous.

Bill: The existence of Hamel bases in (all) vector spaces is equivalent to the axiom of choice (see Blass, Andreas "Existence of bases implies the axiom of choice". Contemporary mathematics 31, 1984). Also, it is consistent with ZF that all linear functionals on $l_2$ are continuous (for instance, a model in which every set of reals has the property of Baire).